Problem 74 - Physics

Question

At provides some pertinent background for this problem. A convertible moves toward you and then passes you; all the while, its loudspeakers are producing a sound. The speed of the car is a constant \(9.00 \mathrm{~m} / \mathrm{s}\), and the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the ratio of the frequency you hear while the car is approaching to the frequency you hear while the car is moving away?

Step-by-Step Solution

Answer

The frequency ratio is approximately 1.0538.

Step 1: Understanding the Doppler Effect

The Doppler Effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. When the source moves towards the observer, the frequency appears to increase; when it moves away, it decreases. We use this concept to solve our problem.

Step 2: Formula for Frequency Ratio

For a stationary observer and moving source, the frequency heard can be calculated using the Doppler Effect formula: \[ f' = \frac{f}{1 - \frac{v_s}{v}}\]when the car is approaching. Conversely, when the car is moving away, the formula is:\[ f'' = \frac{f}{1 + \frac{v_s}{v}}\]where \( f' \) and \( f'' \) are the observed frequencies, \( f \) is the source frequency, \( v_s \) is the speed of the source (9.00 m/s), and \( v \) is the speed of sound (343 m/s).

Step 3: Calculating the Ratio of Frequencies

To find the ratio of the frequency as the car approaches \( (f') \) to when it moves away \( (f'') \), divide the two expressions:\[\text{Ratio} = \frac{f'}{f''} = \frac{\frac{f}{1 - \frac{v_s}{v}}}{\frac{f}{1 + \frac{v_s}{v}}} = \frac{1 + \frac{v_s}{v}}{1 - \frac{v_s}{v}}\]

Step 4: Substitute and Simplify

Substitute \( v_s = 9.00 \mathrm{~m/s} \) and \( v = 343 \mathrm{~m/s} \) into the ratio:\[\text{Ratio} = \frac{1 + \frac{9.00}{343}}{1 - \frac{9.00}{343}} = \frac{1 + 0.0262}{1 - 0.0262}\]which simplifies to:\[\text{Ratio} \approx \frac{1.0262}{0.9738} \approx 1.0538\]

Step 5: Conclusion

The calculated ratio of the frequencies is approximately 1.0538. This means the frequency heard as the car approaches is about 1.0538 times the frequency heard as it moves away.

Key Concepts

Frequency Ratio

The frequency ratio in the context of the Doppler Effect is an important concept to understand. It represents the comparison of sound frequencies heard by an observer when a sound source is moving relative to them. Think of it as a measure of how the pitch of sound changes as the source approaches or recedes.

When the source of sound, like a car with loudspeakers, approaches a stationary observer, the frequency or pitch of the sound increases. Conversely, when the car moves away, the pitch decreases.

• Approaching Source: Frequency seems higher
• Receding Source: Frequency seems lower

For a stationary observer, the frequency ratio (\( \frac{f'}{f''} \)) is calculated using the Doppler Effect formula, where \( f' \) is the frequency heard when the source is approaching, and \( f'' \) is when it is moving away.

Sound Waves

Sound waves are mechanical waves that travel through a medium like air, water, or solids. They are vibrations that our ears interpret as sound. These waves travel in a series of compressions and rarefactions, making them longitudinal waves.

When a source of sound, such as a car, emits these waves while moving, the properties of sound waves, including frequency and wavelength, can change for an observer. This change forms the basis for the Doppler Effect, which explains why sounds seem to vary in pitch when the source moves towards or away from us.

• Compressions: Areas where particles are close together
• Rarefactions: Areas where particles are spread apart

Understanding sound waves is crucial in grasping how they interact with moving objects and change as perceived by an observer.

Observer Moving Relative to Source

For the Doppler Effect, it's essential to consider the movement between the observer and the source of the sound. The entire principle relies on this relative motion.

When the observer moves towards the sound source, or vice versa, the waves are compressed, leading to a higher frequency. If they are moving apart, the waves stretch, leading to a lower frequency. Thus, the relative velocity between the observer and the sound source determines the frequency change.

• Relative Motion: Key to experiencing the Doppler Effect
• Higher frequency: Observer moves towards the source
• Lower frequency: Observer moves away from the source

This movement changes the wave patterns that reach the observer, hence altering the perceived sound.

Speed of Sound

The speed of sound is a fundamental factor in problems related to the Doppler Effect. It represents how fast sound waves travel through a medium. In air, under typical conditions, this speed is approximately 343 m/s.

The formulas to calculate observed frequencies (\( f' \) and \( f'' \)) during relative motion between a sound source and an observer include the speed of sound as a critical parameter. The speed determines how significantly the frequencies are altered by the motion.

• Standard Air Speed: Around 343 m/s
• Influences Doppler Frequency calculations
• Dependent on medium and environmental conditions

Understanding how sound speed plays into equations helps us predict and calculate the changes in sound perception accurately.